9.2: Investigating Special Triangles
This activity is intended to supplement Geometry, Chapter 8, Lesson 4.
ID: 7896
Time required: 45 minutes
Activity Overview
In this activity, students will investigate the properties of an isosceles triangle. Then students will construct a
Topic: Right Triangles & Trigonometric Ratios

Calculate the trigonometric ratios for
45∘−45∘−90∘,60∘−60∘−60∘ and30∘−60∘−90∘ triangles.
Teacher Preparation and Notes
This activity is designed to be used in a high school or middle school geometry classroom.
 If needed, review or introduce the term median of a triangle. Any median of an equilateral triangle is also an altitude, angle bisector, and perpendicular bisector.
 This activity is designed to be studentcentered with the teacher acting as a facilitator while students work cooperatively.
 The worksheet guides students through the main ideas of the activity and provides a place for students to record their work. You may wish to have the class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion.
 To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
 To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=7896 and select EQUIL and ISOSC.
Associated Materials
 Student Worksheet: Investigating Special Right Triangles http://www.ck12.org/flexr/chapter/9693, scroll down to the second activity.
 Cabri Jr. Application
 EQUIL.8xv and ISOSC.8xv
Problem 1 – Investigation of Triangles
First, turn on your TI84 and press APPS. Arrow down until you see Cabri Jr and press ENTER. Open the file ISOSC. This file has a triangle with an isosceles triangle with
Using the Perpendicular tool (ZOOM > Perp.), construct a perpendicular from point
Construct line segments
Would you have expected these segments to be equal in length?
Drag point
Will they always be equal?
Problem 2 – Investigation of 306090 Triangles
Open the file EQUIL. Note that all three angles are
Construct the perpendicular from
From the construction above, we know that
Construct segment
This completes the construction of two
You may choose to have the students hide the segments
Measure the three sides of the triangle.
Press GRAPH and select the Calculate tool. Click on the length of
Drag point
What do you notice about the three ratios?
Problem 3 – Investigation of 454590 Triangles
Press the
To begin the construction of the
Use the compass tool with center
Hide the circle and construct segments
Can you explain why \begin{align*}AB = AC\end{align*} and why angle \begin{align*}ACB = \end{align*} angle \begin{align*}ABC\end{align*}?
Why are these two angles \begin{align*}45^\circ\end{align*} each?
Students should notice that the two angles must be equal, and angle \begin{align*}A\end{align*} is \begin{align*}90^\circ\end{align*}. Therefore, because the sum of the angles in a triangle is \begin{align*}180^\circ\end{align*}, the two angles must be \begin{align*}45^\circ\end{align*} each.
Measure the sides of the triangle. This verifies that \begin{align*}AB = AC\end{align*}.
Use the CALCULATE tool to find the ratio of \begin{align*}AC:BC\end{align*} and \begin{align*}AC:AB\end{align*}. Once again, these ratios will be important when you study trigonometry.
Drag point \begin{align*}B\end{align*} and observe what happens to the sides and ratios.
Why do the ratios remain constant while the sides change?
Students should notice that \begin{align*}AC\end{align*} and \begin{align*}AB\end{align*} are equal, so the ratios will always remain the same.